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Abstract
The focusing spatiotemporal property of a femtosecond laser pulse is presented under tight focusing conditions by using the frequency-resolved incident electric field and vector diffraction formulas with the wavefront correction term. In the ideal case, the focused laser intensity reaches its maximum at the F-number of ∼0.35 due to the strong diffraction effect under extremely tight focusing conditions. In spatio-temporal coupling distortion cases, their spatiotemporal Strehl ratios show a trend of improvement as the F-number decreases and this phenomenon is mainly concentrated along the y-direction. Based on the numerical calculation method used in this work, the precise information of tightly focused ultra-intense femtosecond laser fields can be obtained, which is crucial for assessing a focused intensity and describing the motion of charged particles under an extremely strong electric field. Moreover, the evolution law of focal fields with spatiotemporal distortions found in this paper can offer some theoretical guidance for realizing ultrahigh laser intensity in the near future.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of ultrashort pulse amplification technology in the last two decades, tens of petawatt-class lasers have been constructed worldwide [1]. The ultra-high peak-power laser output of these facilities is widely used in strong-field physics experiments [2]. During laser–matter interactions, the focused peak intensity is a crucial factor in determining the physical processes of strong-field phenomena. Quantum effects, such as vacuum birefringence, can be observed when the intensity exceeds ∼1023 W/cm2 [3,4]. At an intensity close to the Schwinger limit of approximately 1029 W/cm2, a phenomenon known as “strong-field vacuum breakdown” can occur, where electron-positron pairs are directly generated from the vacuum [5,6]. Therefore, to open the door to unknown nature and new science, researchers have been striving to produce higher laser intensity [7–9]. The method for achieving this goal is to pursue nearly perfect laser beams and extremely tight focusing conditions. For the former, the full spatiotemporal structure of ultra-intense femtosecond laser beams has attracted much attention in recent years [10–14]. Multi-PW lasers are susceptible to various factors, such as lens chromatic aberration and grating defects, resulting in significant spatio-temporal couplings (STCs) of the light field. This implies considerable differences in the temporal properties of the laser beam at different spatial positions. Thus, STCs typically increase the pulse duration and reduce peak intensity in focus, which causes the traditional method of assessing the peak focus intensity using independent spatial and temporal measurements to be no longer accurate [15]. As for the tight focusing conditions, scientists usually choose paraboloidal mirrors with small F-numbers to satisfy the requirements [7–9,16]. The characterization of tightly focused vector fields formed by paraboloidal mirrors has been extensively studied [17–24]. Most studies on tightly focused laser fields have not considered or analyzed the impact of STCs, and a few papers have reported some simple investigations on the spatiotemporal intensity distribution of a focused pulse under the influence of lower-order STCs [25]. However, there is no in-depth study on the evolution law of the tightly focused spatiotemporal properties with decreasing F-number, from 1 to 0.25, of a paraboloidal mirror in the STCs cases (especially in the complex higher-order STC case). This gap presents challenges for researchers to further precisely speculate and depict the practical information on tightly focused high-power femtosecond laser fields under the STCs conditions, which is crucial for simulating and predicting the motion of charged particles in ultra-strong electromagnetic fields.
In this paper, by introducing STCs modification, we investigate the spatiotemporal characterization of a femtosecond laser pulse under extremely tight focusing conditions (below F-number of 1) for the first time. In Section 2, a mathematical model for describing the spatiotemporal coupled electric field distribution focused with low F-number paraboloidal mirrors is developed at first. Besides, we propose the concept of “the effective spatiotemporal volume” to accurately calculate the focused peak intensity of femtosecond laser beams with STCs. In Section 3, numerical simulation results for focal spots with various STCs and different F-number optics, including tightly focused vector field component spatiotemporal profiles, the spatiotemporal Strehl ratio, the spatiotemporal effective pulse duration, etc., is presented and discussed. Finally, the summary is made in section 4. The method described in this paper is applicable for the precise evaluation of the tightly focused spatiotemporal property of a high-power femtosecond laser pulse. Furthermore, the evolution law of spatiotemporal characteristics on the tightly focused femtosecond laser field with decreasing F-number of a focusing optic revealed in this work can provide theoretical guidance for the realization of ultrahigh laser intensity in the near future.
2. Mathematical modeling
2.1 Vector diffraction formulas for a paraboloidal mirror
In order to calculate the polychromatic electric vector field focused by a paraboloidal mirror, a linearly-polarized (polarized in the x-axis), free-propagating, femtosecond laser beam incident on the paraboloidal mirror with a focal length of f from the right is considered (see Fig. 1). The broadband femtosecond laser pulse can be divided into a sequence of monochromatic electromagnetic waves having their own amplitude and phase distributions. Thus, the incident electric field Einc(x,y,ω) can be given by:
here, the amplitude is considered to have no spatio-spectral coupling and can be expressed as: where S(ω) is the normalized spectrum of the femtosecond laser, As(x,y) is the normalized spatial intensity distribution. And the spatio-spectral phase is given by [15]:
Fig. 1. On-axis focusing model for a femtosecond laser beam with a low F-number paraboloidal mirror.
According to [16], by setting the rotation angle φ and the distance h from the z-axis to the center of the incident beam to 0, the vector diffraction formulas near the focus of an on-axis paraboloidal mirror in the Cartesian coordinate system can be written as:
Based on Eqs. (1)-(10), the tightly focused electric vector field E(xf,yf,ω) of a broadband femtosecond laser on the focal plane can be obtained. Then, the spatiotemporal electric field E(xf,yf,t) of the focused pulse can be calculated by the inverse Fourier transform of E(xf,yf,ω) in the frequency domain.
2.2 Implementation of STCs and the concept of “the effective spatiotemporal volume”
In this paper, we investigate the tightly focused femtosecond laser field under the influence of three types of STCs separately, which are pulse-front tilt (PFT), pulse-front curvature (PFC), and the complex spatio-temporal coupling induced by wavefront errors in a grating compressor (GC-CSTC). PFT and PFC are the most prevalent and low-order STCs, which are caused by very simple and common optical elements. According to [15], the spectrally-resolved spatial phase distributions of a broadband femtosecond laser can be described by the frequency-domain extension of Zernike polynomials, which is given by:
GC-CSTC has been intensively studied by Zhaoyang Li et al [26,27]. Based on Eqs. (1)-(5) in the Ref. [26], the spectrally-resolved spatial phase for GC-CSTC can be simulated:
Under the influence of STCs, the traditional method for calculating the focused peak intensity based on the focal spot area and pulse duration will significantly overestimate the actual laser intensity. To solve this problem, considering the inseparability of time and space in STCs, we propose the concept of “the effective spatiotemporal volume”, which is given by:
where In(x,y,t) is the normalized intensity distribution in the 3D space-time domain. Since the pulse energy $J = \int\!\!\!\int\!\!\!\int {I(x,y,t)} dxdydt = {I_{peak}}\int\!\!\!\int\!\!\!\int {{I_n}(x,y,t)} dxdydt$, the peak laser intensity is: According to [7,12], the effective pulse duration and the effective spot area in the 3D space-time domain can be given by:2.3 Simulation parameters
The spectrum and the spectral phase of a femtosecond laser pulse used in our model are shown in Fig. 2. The data are taken from a measurement using the Wizzler in the Shanghai Super-intense Ultrafast Laser Facility (SULF) [28]. In our model, the number of frequency samples is 50. For every specific frequency ω, a circular x-polarized electric field with a Gaussian beam profile is used as its incident field, and the beam radius is 250 mm. The spatial resolutions of the incident laser field and the tightly focused field are 10 mm and 0.1 µm, respectively. The simulation of PFT is chosen along the x direction. For GC-CSTC, the total chirp rate, the grating groove density, the low/high spatial frequency wavefront error of gratings, and other related parameters are the same as those in [26].
3. Numerical simulation results and analysis
3.1 Tightly focused ideal femtosecond laser fields
In this section, by setting the spatio-spectral phase ϕ(x,y,ω) of the femtosecond laser pulse to 0, the spatiotemporal characterization of tightly focused ideal optical fields is calculated. Firstly, ${I_x} = {|{{E_x}({x_f},{y_f},t)} |^2}$, ${I_y} = {|{{E_y}({x_f},{y_f},t)} |^2}$, ${I_z} = {|{{E_z}({x_f},{y_f},t)} |^2}$ and $I = {I_x} + {I_y} + {I_z}$ are plotted as intensity distributions in the 3D space-time domain for x-, y-, z-polarized and entire electric fields, respectively. Figure 3 shows the change of a focal spot for an ideal electric field without STC as the F-number decreases. It can be intuitively seen that although the main lobe size of the focal spot shrinks gradually as the F-number decreases, the intensity of its diffraction sidelobe increases, indicating that the diffraction effect is becoming increasingly severe. Moreover, when the F-number is 0.35 and 0.25, there is little change in the size of the focal spot's main lobe, but the intensity of its diffraction sidelobes noticeably grows, demonstrating a reduction in the encircled energy at the focus. Meanwhile, we also separately calculated the ratios of peak intensities of the longitudinal electric field component (Ez) and the y-polarized field component (Ey) to that of the x-polarized field component (Ex). As shown in Fig. 3, as the F-number decreases from 2 to 0.25, the peak intensity of Ey increases from almost 0% to 7.1% of that of Ex, while the peak intensity of Ez increases from ∼0.6% to 50% of that of Ex. This indicates that under low F-number conditions (especially below 1), both Ey and Ez affect the overall electric field intensity distribution. The latter's impact is predominant, which significantly changes the entire intensity profile into an obround shape.
Fig. 3. Intensity distributions in the 3D space-time domain for an ideal femtosecond laser beam under various focusing conditions. In the figure, the left axis on the horizontal plane is the y-axis and the right is the x-axis (unit: µm). The vertical axis is the time axis (unit: fs). On the time = −100 fs, x = 5/2.5 µm and y = 5/2.5 µm three planes, the intensity contour maps (0.1Imax∼Imax) of time = 0, x = 0 and y = 0 are drawn, respectively.
To further investigate the evolution law of focus spatiotemporal characteristics under different tight focusing conditions, the ratios Iy,z_max/Ix_max, the effective pulse duration, the effective spot area and the effective spatiotemporal volume vs. the F-number are plotted in Fig. 4. As shown in Fig. 4(a), the ratios Iy,z_max/Ix_max increases exponentially with the F-number reduction (the trend of change in the ratio Iz_max/Ix_max is readily apparent in the linear coordinate system, whereas the ratio Iy_max/Ix_max exhibits insignificant changes due to its negligible value when F/#>0.5). Moreover, it should be noticed that the case of F/#=0.25 (i.e. NA = 1, which means the peripheral beam is incident on the focus at the right angle of π/2) is a critical state. Under this F-number condition, the sudden increase in ratios Iy,z_max/Ix_max is observed, and the same phenomenon is also observed for the effective pulse duration, the effective spot area and the effective spatiotemporal volume. From Fig. 4(b), we can see that the value of $T^{ST}_{eff}$ significantly increase when F/#<0.5 due to the increasingly severe diffraction effect. Specifically, under extremely tight focusing conditions, the strong diffraction enhances spatial intensity integration $\int\!\!\!\int {I(x,y,{t_{edge}})dxdy}$ of the femtosecond pulse edge, which ultimately causes the spatiotemporal effective pulse duration to increase. The strong diffraction also causes $A^{ST}_{eff}$ and $V^{ST}_{eff}$ to have a minimum value at F/#=∼0.35 (see Fig. 4(c) and (d)), which means that the focused laser intensity reaches its maximum value under this condition according to the Eq. (16).
3.2 Tightly focused femtosecond laser fields with STC distortions
In this section, we simulate the tightly focused femtosecond laser field with three types of STCs distortions separately. Firstly, the influence of PFT (γx = 0.2 fs/mm) is investigated. As shown in Fig. 5, with the decrease of F-numbers, the distortion of the spatiotemporal intensity profile caused by PFT becomes gradually weaker. Meanwhile, the diffraction effect is also becoming increasingly severe, as in the ideal case.
Fig. 4. Under the ideal condition, (a) The ratios Iy_max/Ix_max and Iz_max/Ix_max, (b) the effective pulse duration, (c) the effective spot area, (d) the effective spatiotemporal volume, as a function of the F-number.
Fig. 5. Intensity distributions in the 3D space-time domain for a femtosecond laser beam with a PFT distortion under various focusing conditions. The figure attribute parameters are the same as Fig. 3.
Based on the data presented in Fig. 5 and Fig. 6(a), we observe that the ratios Iy,z_max/Ix_max are slightly larger than those obtained under ideal conditions. However, the exponential increase trend remains consistent with the ideal case. The above phenomenon indicates the PFT can enhance the longitudinal and the y-polarized electric field components. Figure 6(b) shows that the value of $T^{ST}_{eff}$ experiences a decline followed by an increase when the F-numbers gradually decrease. The decline phenomenon demonstrates a gradual reduction in the distortion caused by PFT, while the increasing phenomenon suggests the opposite. Moreover, the diffraction effect also plays a key role when the value of $T^{ST}_{eff}$ increases. These factors result in the minimum value of $T^{ST}_{eff}$ at F/#=∼0.35. As depicted in Fig. 6(c) and (d), the distortion caused by PFT leads to larger values of $A^{ST}_{eff}$ and $V^{ST}_{eff}$ compared to those obtained under ideal conditions. Nevertheless, their changing trends remain consistent with the ideal case and their minimum values also occur at F/#=∼0.35.
Fig. 6. Under the PFT condition, (a) The ratios Iy_max/Ix_max and Iz_max/Ix_max, (b) the effective pulse duration, (c) the effective spot area, (d) the effective spatiotemporal volume, as a function of the F-number.
Secondly, we study the influence of PFC (α=0.001 fs/mm2), and the simulation results are shown in Fig. 7 and Fig. 8. It can be seen from Fig. 7 that, with the decrease of F-numbers, the distortion of the spatiotemporal intensity profile caused by PFC gradually reduces and then increases again. When F/#=∼0.5, the degree of distortion reaches a minimum. Moreover, we can also observe that the distortion varies more significantly in the y direction than in the x direction when F-numbers decrease. For example, in the spatiotemporal intensity profile changes of Ex, the distorted shape in the y direction changes from “▴” to “▪” and then to “▾”, while the distorted shape in the x direction remains as “▴” throughout. Meanwhile, like the ideal and PFC cases, the diffraction effect also becomes increasingly severe.
Fig. 7. Intensity distributions in the 3D space-time domain for a femtosecond laser beam with a PFC distortion under various focusing conditions. The figure attribute parameters are the same as Fig. 3.
Fig. 8. Under the PFC condition, (a) The ratios Iy_max/Ix_max and Iz_max/Ix_max, (b) the effective pulse duration, (c) the effective spot area, (d) the effective spatiotemporal volume, as a function of the F-number.
In contrast to the PFT case, the ratios Iy,z_max/Ix_max in the PFC case are slightly smaller than those obtained under ideal conditions (see Fig. 7 and Fig. 8(a)). However, the exponential increase trend still remains consistent with the ideal case. The above phenomenon means that the PFT can reduce the longitudinal and the y-polarized electric field components. Besides, as shown in Fig. 8(b), (c) and (d), the changing trends of $T^{ST}_{eff}$, $A^{ST}_{eff}$ and $V^{ST}_{eff}$ in the PFC case are the same as those in the PFT case. Here, the minimum value of $T^{ST}_{eff}$ occurs at F/#=∼0.55, while the minimum values of $A^{ST}_{eff}$ and $V^{ST}_{eff}$ still occur at F/#=∼0.35.
Finally, the influence of the GC-CSTC on the tightly focused femtosecond laser field is simulated. Figure 9 shows that the GC-CSTC results in several side lobes appearing in the focal field, and the distortion gradually weakens as the F-number decreases. The phenomenon of “distortion reduction” observed here is similar to that seen in the PFC and PFT cases. And like in other cases, the diffraction effect is also becoming increasingly severe. Besides, as shown in Fig. 9 and Fig. 10(a), the ratios Iy,z_max/Ix_max are larger than those obtained under ideal conditions and still have an exponential increase trend as the F-number decreases, which indicates that the GC-CSTC can enhance Ey and Ez components. The changing trends of $T^{ST}_{eff}$, $A^{ST}_{eff}$ and $V^{ST}_{eff}$ in the GC-CSTC case are the same as those in the first two cases (see Fig. 10(b), (c) and (d)). Here, the minimum values of $T^{ST}_{eff}$, $A^{ST}_{eff}$ and $V^{ST}_{eff}$ all occur at around F/#=0.35.
Fig. 9. Intensity distributions in the 3D space-time domain for a femtosecond laser beam with a GC-CSTC distortion under various focusing conditions. The figure attribute parameters are the same as Fig. 3.
Fig. 10. Under the GC-CSTC condition, (a) The ratios Iy_max/Ix_max and Iz_max/Ix_max, (b) the effective pulse duration, (c) the effective spot area, (d) the effective spatiotemporal volume, as a function of the F-number.
In order to further investigate the phenomenon of “distortion reduction”, we calculate the spatiotemporal Strehl ratio SRSTC [15] as a function of the F-number in the PFT, PFC and GC-CSTC cases. It can be seen from Fig. 11(a) that under the PFT and PFC conditions, the SRSTC has a maximum value at F/#=∼0.28 (near the critical value of 0.25) and F/#=∼0.55, respectively. However, the SRSTC of the GC-CSTC keeps increasing as the F-number decreases. In any case, there is indeed the phenomenon of “distortion reduction” under low F-number parabolic mirror focusing. This phenomenon can be mainly attributed to the correction term for wavefront aberration caused by polarization rotation after reflection from a mirror surface [19] (see the [•] term in the Eq. (7)). The numerator of the subtrahend in the [•] term only contains the variable y, which means the correction is mainly concentrated along the y-direction. This conclusion is also confirmed by the more significant variation of distortion observed in the y-direction under the PFC condition. To more intuitively observe the influence of the correction term, we set the uniform spatial phase of the incident laser field to 1 rad and simulate the variation of the corrected phase with the F-number. As shown in Fig. 11(b), the variation of spatial phase is not apparent when F/#>1. However, when the F-number decreases from 1 to 0.25, the spatial phase on both sides of the light spot along the y-direction changes sharply from about 0.6 rad to −1 rad, and the spatial phase along the x-direction (especially near the center of the light spot) still remains at ∼1 rad. This indicates that the wavefront aberration of the light spot tends to decrease first and then increase when F/#<1, and this change is more significant in the y-direction. Finally, through the calculation of vector diffraction formulas, the distortion of the focal spot will also exhibit a similar trend.
4. Conclusion
By using the frequency-resolved incident electric field and vector diffraction formulas with considering the correction term for wavefront aberration, a numerical model for calculating spatiotemporal characteristics of the tightly focused femtosecond laser field has been developed. Based on this model, we investigate the variations of tightly focused vector field component spatiotemporal intensity profiles, the spatiotemporal effective pulse duration $T^{ST}_{eff}$ and spot area $A^{ST}_{eff}$, the effective spatiotemporal volume $V^{ST}_{eff}$, and the spatiotemporal Strehl ratio SRSTC with the F-number under the ideal, PFT, PFC and GC-CSTC conditions. The calculation results show that due to the strong diffraction effect, even under the ideal condition, the focused laser intensity will reach its maximum at F/#=∼0.35. Besides, under the assumption that the incident light field is x-polarized, the longitudinal electric field component Ez increases exponentially with the F-number reduction in all cases and accounts for a larger proportion in the PFT and GC-CSTC cases. For all STCs, due to the wavefront correction term caused by polarization rotation after reflection from a paraboloidal mirror surface, a phenomenon known as “distortion reduction” can be observed under low F-number conditions, which means their spatiotemporal Strehl ratios show a trend of improvement. Moreover, the influence of this wavefront correction term is mainly concentrated along the y-direction. In conclusion, the numerical calculation model used in this paper and the concept of “the effective spatiotemporal volume” can accurately characterize a femtosecond focal spot and its focused peak intensity, especially under the influence of STCs, which is essential for obtaining realistic information on extreme intensities formed by tightly focused ultra-intense laser pulses. Furthermore, the evolution law of STC distortions on the focal field under low F-number conditions revealed in this work can provide some theoretical guidance towards achieving ultrahigh laser intensity shortly.
Fig. 11. (a) Under the PFT, PFC and GC-CSTC conditions, the spatiotemporal Strehl ratio SRSTC versus the F-number. (b) the local section view of the corrected spatial phase as a function of F-number.
Funding
Program of Shanghai Academic Research Leader (20SR014501); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019247).
Acknowledgment
This work was supported by the Zhangjiang Laboratory.
Disclosures
The authors declare no conflicts of interest.
Data availability
All models and data presented in this study are available from the corresponding author upon reasonable request.
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